“A Transantiago bus carries 34 passengers. At a bus stop, 12 get off and 17 get on. How many passengers continue up to the next bus stops? ”
“The last digit of the Juanita’s numbers is the sum of the other digits. For example, the number 2013 is a number of Juanita. Then, what is the largest number of Juanita not having zero as a digit ? ”
Since 2012, Chilean teachers should devote more time to ask their students for math problems like these. Since then, problems solving was no longer just a content and became an explicit ability to be developed in the curriculum, along with three other abilities: arguing and communicating mathematics, representing and modeling. This sets teachers, school directors and trainers of future teachers a new challenge, especially since problem solving seems to be absent from initial formation and school classrooms. “There is not a broad base of research to say this, but there are many indirect evidence that it is the case. First, the result of our students in international tests such as PISA, where problem solving has a significant impact, here our students show weak results,” says the National Prize of Sciences laureate and researcher of the Center for Mathematical Modeling and the Center for Advanced Research Education (CIAE), Patricio Felmer.
A study led by Felmer –and funded by the Fund for Research and Development in Education (Fonide) of the Ministry of Education– analyzed a sample of thirty secondary teachers graduated less than three years ago and detected that the majority of teachers surveyed had little or no opportunity to work problem solving in their initial training. This leads them to offer few problem solving opportunities to their students in classroom.
“At the university, problem solving is very briefly seen. It does not look like a unit in itself,” says Noemí, an elementary teacher at Guillermo Matta School of SIP, who teaches math to third grade children. Patricia has the same opinion. She teaches in a municipal elementary school in Cerrillos, she has an experience of 20 years doing math classes and now is teaching in fourth grade of an elementary school. “I just received a brief teaching in the university and I have learnt by doing. I developed mathematics by doing,” she said. This situation is changing. In schools, there is a particular interest from holders and heads of technical unit in education (UTP, in spanish) that teachers incorporate problem solving as an everyday practice. “Today’s society requires citizens who develop productive habits of mind and are able to ask questions about different situations and to develop strategies to solve them. From this point of view, schools should be responsible to take this challenge by incorporating problems solving from different perspectives, not only from mathematics,” says Marcia Muñoz Varisco, mathematics coordinator from SIP Schools Network.
Defining moment
“What is a mathematical problem?,” CIAE and CMM researcher, Josefa Perdomo asks 20 elementary teachers in a RPAula workshop developed by both institutions of the University of Chile. “It’s a challenge,” says a teacher. “For children, it is a game,” says another teacher. Thus, teachers, along with the researcher, are agreeing on what is the most accurate definition of a math problem. “A mathematical activity is a problem when you do not know what is the procedure that takes you to the solution” says Josefa Perdomo, who adds other elements: it is not necessarily to be linked to the context and it depends on the level of knowledge of the person facing the problem, therefore, a mathematical problem will not be equal for a second grade elementary student or for a second grade high school student. So when the teacher gives an activity that has a claer way to be solved given or given by the teacher, it is no longer a mathematical problem, but an exercise. Furthermore, when the student solves the problem, it ceases to be a problem for him.
Watch ABC problem solving
“Teachers usually teach only one way to solve problems. Here, they teach us many different ways,”says Yennifer, a math teacher from a municipal school in Cerrillos. A reality that Felmer’s study detected when analyzing how the effective time during class was distributed by high school teachers with less than three years of experience. 35% of time, the teacher delivers solutions, avoiding students to search for themselves. The activities of the “Professor promoting discussion” or “a student asking a non-routine question” are given in very few cases (less than 3% of the class), while the “teacher returns the responsibility ” is recorded on a larger percentage but still very low (10%).
“In addition to time, getting students to discover and question fundamental ideas of the discipline requires knowing how to do it. Planning what the right questions and not delivering the solution to the problems, but to answer questions with new questions, is something not seen very often in the classroom,” says Marcia Muñoz.
Lack of time
Every day, early in her class, Noemí states a problem for third grade elementary students, related to the unit they are studying. “Kids love it and are highly motivated,” she says, but faces the obstacle with all teachers: lack of time. “Children don’t use five minutes to solve it, because a problem opens the door to other questions,” she says. Marcia Muñoz agrees: “When we know that students should acquire knowledge to properly answer a Simce or PSU test, you must first ensure curriculum coverage. Although we make the best efforts to incorporate the development of skills and strategies to solve problems, we finally know times to achieve both tasks are reduced.” Closely linked to the lack of time is the size of the classes that prevents teachers to adequately address individual questions of students. “It’s harder to mediate a group of 45 students than a course of 30. Although they are organized in groups and I assume the role of mediator in the classroom, I keep having the feeling of not attending the needs of all students, we tend to fall in the expository model of classes,” explains the mathematics coordinator of SIP
Network
Children understanding of the problems is another obstacle faced by teachers: children must read and understand the statements, which is not easy at all. “The basic problem solving skills are reading comprehension, writing and the ability to abstract. It is not only calculation. And often children have not strengthened these skills. In response, students are often frustrated,” says Patricia.
For Perdomo, this apparent disadvantage can be turned into an opportunity, because problem solving activities help precisely to develop the other three skills: “To problem solving is necessary to represent, model and argue and communicate. Problem solving provides excellent opportunities for communication: teamwork, communication with the teacher and discussions where students present their strategies,” says the researcher.
ARPA
It’s Tuesday afternoon and about 20 teachers from the SIP School Network and Fundación Astoreca have spent the first half of the day solving problems on their own. Soon, they discuss what is meant by a mathematical problem, to be prepared to “activate problem solving skills” in their own courses.
Why “activate?” Problem solving skill is an innate ability in human beings. One-year old children solve probelma and our ancestors also solved problems, to meet different needs, such as hunting, for example. We want to enhance this innate skill in teachers and students and eliminate false beliefs such as, for example, to solve a problem you have to know a formula,” says Felmer.
That is precisely the goal of the ARPA Initiative, acronym of Activando la Resolución de Problemas en las Aulas (Activating Problem Solving in Classrooms): generating strategies for teacher professional training aiming to activate the problem solving in classrooms.
The ARPA initiative is developed under the umbrella of the CMM and CIAE and this year is planning to impact more than a thousand primary school teachers and also student of Education Faculties. “The initiative seeks to promote math skills in the classroom, enriching mathematics, and making children experience it in all its dimensions. This, by proposing professional development strategies for teachers, for whom their students’ learning is their passion,” says Felmer.